The concept of limits forms the basis of calculus and is a very powerful one. Both differential and integral calculus are based on this concept and as such, limits need to be studied in good detail.
This section contains a general, intuitive introduction to limits.
Consider a circle of radius .
We know that the area of this circle is . How
The ancient Greeks derived this result using the concept of limits.
To see how, recall the definition of .
With this definition in hand, the Greeks divided the circle as follows (like cutting a cake or a pie):
Now they took the different pieces of this ‘pie’ and placed them as follows:
See what happens if the number of cuts are increased
The figure on the right side starts resembling a rectangle as we increase the number of cuts to the circle. The sequence of curves that joins to starts becoming more and more of a straight line with the same total length .
What happens as we increase the number of cuts indefinitely, or equivalently, we decrease indefinitely? The figure ‘almost’ becomes a rectangle, though never becoming a rectangle exactly. The area ‘almost’ becomes
In the language of limits, we say that the figure tends to a rectangle or the area tends to or the limiting value of area is